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<a href="decompositions_8hpp.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno">    1</span><span class="comment">/*!</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno">    2</span><span class="comment"> * \brief       Defines types for matrix decompositions and solvers</span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno">    3</span><span class="comment"> * </span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno">    4</span><span class="comment"> * \author      O. Krause</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno">    5</span><span class="comment"> * \date        2016</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno">    6</span><span class="comment"> *</span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno">    7</span><span class="comment"> *</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno">    8</span><span class="comment"> * \par Copyright 1995-2015 Shark Development Team</span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno">    9</span><span class="comment"> * </span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno">   10</span><span class="comment"> * &lt;BR&gt;&lt;HR&gt;</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno">   11</span><span class="comment"> * This file is part of Shark.</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno">   12</span><span class="comment"> * &lt;http://image.diku.dk/shark/&gt;</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno">   13</span><span class="comment"> * </span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno">   14</span><span class="comment"> * Shark is free software: you can redistribute it and/or modify</span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno">   15</span><span class="comment"> * it under the terms of the GNU Lesser General Public License as published </span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno">   16</span><span class="comment"> * by the Free Software Foundation, either version 3 of the License, or</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno">   17</span><span class="comment"> * (at your option) any later version.</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno">   18</span><span class="comment"> * </span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno">   19</span><span class="comment"> * Shark is distributed in the hope that it will be useful,</span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno">   20</span><span class="comment"> * but WITHOUT ANY WARRANTY; without even the implied warranty of</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno">   21</span><span class="comment"> * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno">   22</span><span class="comment"> * GNU Lesser General Public License for more details.</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno">   23</span><span class="comment"> * </span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno">   24</span><span class="comment"> * You should have received a copy of the GNU Lesser General Public License</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno">   25</span><span class="comment"> * along with Shark.  If not, see &lt;http://www.gnu.org/licenses/&gt;.</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno">   26</span><span class="comment"> *</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno">   27</span><span class="comment"> */</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno">   28</span><span class="preprocessor">#ifndef REMORA_DECOMPOSITIONS_HPP</span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno">   29</span><span class="preprocessor">#define REMORA_DECOMPOSITIONS_HPP</span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno">   30</span> </div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno">   31</span><span class="preprocessor">#include &quot;<a class="code" href="trsm_8hpp.html">kernels/trsm.hpp</a>&quot;</span></div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno">   32</span><span class="preprocessor">#include &quot;<a class="code" href="trsv_8hpp.html">kernels/trsv.hpp</a>&quot;</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno">   33</span><span class="preprocessor">#include &quot;<a class="code" href="potrf_8hpp.html">kernels/potrf.hpp</a>&quot;</span></div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno">   34</span><span class="preprocessor">#include &quot;<a class="code" href="pstrf_8hpp.html">kernels/pstrf.hpp</a>&quot;</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno">   35</span><span class="preprocessor">#include &quot;<a class="code" href="getrf_8hpp.html">kernels/getrf.hpp</a>&quot;</span></div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno">   36</span><span class="preprocessor">#include &quot;<a class="code" href="syev_8hpp.html">kernels/syev.hpp</a>&quot;</span></div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno">   37</span><span class="preprocessor">#include &quot;<a class="code" href="assignment_8hpp.html">assignment.hpp</a>&quot;</span></div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno">   38</span><span class="preprocessor">#include &quot;<a class="code" href="permutation_8hpp.html">permutation.hpp</a>&quot;</span></div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno">   39</span><span class="preprocessor">#include &quot;<a class="code" href="matrix__expression_8hpp.html">matrix_expression.hpp</a>&quot;</span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno">   40</span><span class="preprocessor">#include &quot;<a class="code" href="proxy__expressions_8hpp.html">proxy_expressions.hpp</a>&quot;</span></div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno">   41</span><span class="preprocessor">#include &quot;<a class="code" href="vector__expression_8hpp.html">vector_expression.hpp</a>&quot;</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno">   42</span> </div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno">   43</span><span class="keyword">namespace </span>remora{</div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno">   44</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> D, <span class="keyword">class</span> Device&gt;</div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno">   45</span><span class="keyword">struct </span>solver_expression{</div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno">   46</span>    <span class="keyword">typedef</span> Device device_type;</div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno">   47</span>    </div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno">   48</span>    D <span class="keyword">const</span>&amp; operator()()<span class="keyword"> const </span>{</div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno">   49</span>        <span class="keywordflow">return</span> *<span class="keyword">static_cast&lt;</span>D const*<span class="keyword">&gt;</span>(<span class="keyword">this</span>);</div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno">   50</span>    }</div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno">   51</span> </div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno">   52</span>    D&amp; operator()() {</div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno">   53</span>        <span class="keywordflow">return</span> *<span class="keyword">static_cast&lt;</span>D*<span class="keyword">&gt;</span>(<span class="keyword">this</span>);</div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno">   54</span>    }</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno">   55</span>};</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno">   56</span> </div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno">   57</span><span class="comment"></span> </div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno">   58</span><span class="comment">/// \brief Lower triangular Cholesky decomposition.</span></div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno">   59</span><span class="comment">///</span></div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno">   60</span><span class="comment">///  Given an \f$ m \times m \f$ symmetric positive definite matrix</span></div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno">   61</span><span class="comment">///  \f$A\f$, represents the lower triangular matrix \f$L\f$ such that</span></div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno">   62</span><span class="comment">///  \f$A = LL^T \f$.</span></div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno">   63</span><span class="comment">///  </span></div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno">   64</span><span class="comment">/// This decomposition is a corner block of many linear algebra routines</span></div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno">   65</span><span class="comment">/// especially for solving symmetric positive definite systems of equations.</span></div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno">   66</span><span class="comment">/// </span></div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno">   67</span><span class="comment">/// Unlike many other decompositions, this decomposition is fast,</span></div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno">   68</span><span class="comment">/// numerically stable and can be updated when the original matrix is changed</span></div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno">   69</span><span class="comment">/// (rank-k updates of the form A&lt;-alpha A + VCV^T)</span></div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno">   70</span><span class="comment"></span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatrixStorage&gt;</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno">   71</span><span class="keyword">class </span>cholesky_decomposition:</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno">   72</span>    <span class="keyword">public</span> solver_expression&lt;</div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno">   73</span>        cholesky_decomposition&lt;MatrixStorage&gt;, </div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno">   74</span>        typename MatrixStorage::device_type</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno">   75</span>&gt;{</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno">   76</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno">   77</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixStorage::value_type value_type;</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno">   78</span><span class="keyword">public</span>:</div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno">   79</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixStorage::device_type device_type;</div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno">   80</span>    cholesky_decomposition(){}</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno">   81</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> E&gt;</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno">   82</span>    cholesky_decomposition(matrix_expression&lt;E,device_type&gt; <span class="keyword">const</span>&amp; e):m_cholesky(e){</div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno">   83</span>        kernels::potrf&lt;lower&gt;(m_cholesky);</div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno">   84</span>    }</div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno">   85</span>    </div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno">   86</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> E&gt;</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno">   87</span>    <span class="keywordtype">void</span> decompose(matrix_expression&lt;E,device_type&gt; <span class="keyword">const</span>&amp; e){</div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno">   88</span>        m_cholesky.resize(e().size1(), e().size2());</div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno">   89</span>        noalias(m_cholesky) = e;</div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno">   90</span>        kernels::potrf&lt;lower&gt;(m_cholesky);</div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno">   91</span>    }</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno">   92</span>    </div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno">   93</span>    MatrixStorage <span class="keyword">const</span>&amp; lower_factor()<span class="keyword">const</span>{</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno">   94</span>        <span class="keywordflow">return</span> m_cholesky;</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno">   95</span>    }</div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno">   96</span> </div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno">   97</span>    <span class="keyword">auto</span> upper_factor()const -&gt; decltype(trans(std::declval&lt;MatrixStorage const&amp;&gt;())){</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno">   98</span>        <span class="keywordflow">return</span> trans(m_cholesky);</div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno">   99</span>    }</div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno">  100</span>    </div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno">  101</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno">  102</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, left)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno">  103</span>        kernels::trsm&lt;lower,left &gt;(lower_factor(),B);</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno">  104</span>        kernels::trsm&lt;upper,left &gt;(upper_factor(),B);</div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno">  105</span>    }</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno">  106</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno">  107</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, right)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno">  108</span>        kernels::trsm&lt;upper,right &gt;(upper_factor(),B);</div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno">  109</span>        kernels::trsm&lt;lower,right &gt;(lower_factor(),B);</div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno">  110</span>    }</div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno">  111</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB, <span class="keywordtype">bool</span> Left&gt;</div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno">  112</span>    <span class="keywordtype">void</span> solve(vector_expression&lt;VecB,device_type&gt;&amp;b, system_tag&lt;Left&gt;)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno">  113</span>        kernels::trsv&lt;lower, left&gt;(lower_factor(),b);</div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno">  114</span>        kernels::trsv&lt;upper,left&gt;(upper_factor(),b);</div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno">  115</span>    }</div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno">  116</span>    <span class="comment"></span></div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno">  117</span><span class="comment">    /// \brief Updates a covariance factor by a rank one update</span></div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno">  118</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno">  119</span><span class="comment">    /// Let \f$ A=LL^T \f$ be a matrix with its lower cholesky factor. Assume we want to update </span></div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno">  120</span><span class="comment">    /// A using a simple rank-one update \f$ A = \alpha A+ \beta vv^T \f$. This invalidates L and</span></div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno">  121</span><span class="comment">    /// it needs to be recomputed which is O(n^3). instead we can update the factorisation</span></div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno">  122</span><span class="comment">    /// directly by performing a similar, albeit more complex algorithm on L, which can be done</span></div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno">  123</span><span class="comment">    /// in O(L^2). </span></div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno">  124</span><span class="comment">    /// </span></div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno">  125</span><span class="comment">    /// Alpha is not required to be positive, but if it is not negative, one has to be carefull</span></div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno">  126</span><span class="comment">    /// that the update would keep A positive definite. Otherwise the decomposition does not</span></div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno">  127</span><span class="comment">    /// exist anymore and an exception is thrown.</span></div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno">  128</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno">  129</span><span class="comment">    /// \param v the update vector</span></div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno">  130</span><span class="comment">    /// \param alpha the scaling factor, must be positive.</span></div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno">  131</span><span class="comment">    /// \param beta the update factor. it Can be positive or negative</span></div>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno">  132</span><span class="comment"></span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecV&gt;</div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno">  133</span>    <span class="keywordtype">void</span> update(value_type alpha, value_type beta, vector_expression&lt;VecV,device_type&gt; <span class="keyword">const</span>&amp; v){</div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno">  134</span>        <span class="keywordflow">if</span>(beta == 0){</div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno">  135</span>            m_cholesky *= std::sqrt(alpha);</div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno">  136</span>            <span class="keywordflow">return</span>;</div>
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno">  137</span>        }</div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno">  138</span>        <span class="comment">//implementation blatantly stolen from Eigen</span></div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno">  139</span>        std::size_t n = v().size();</div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno">  140</span>        <span class="keyword">auto</span>&amp; L = m_cholesky;</div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno">  141</span>        <span class="keyword">typename</span> vector_temporary&lt;VecV&gt;::type temp = v;</div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno">  142</span>        <span class="keywordtype">double</span> beta_prime = 1;</div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno">  143</span>        <span class="keywordtype">double</span> a = std::sqrt(alpha);</div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno">  144</span>        <span class="keywordflow">for</span>(std::size_t j=0; j != n; ++j)</div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno">  145</span>        {</div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno">  146</span>            <span class="keywordtype">double</span> Ljj = a * L(j,j);</div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno">  147</span>            <span class="keywordtype">double</span> dj = Ljj * Ljj;</div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno">  148</span>            <span class="keywordtype">double</span> wj = temp(j);</div>
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno">  149</span>            <span class="keywordtype">double</span> swj2 = beta * wj * wj;</div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno">  150</span>            <span class="keywordtype">double</span> gamma = dj * beta_prime + swj2;</div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno">  151</span> </div>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno">  152</span>            <span class="keywordtype">double</span> x = dj + swj2/beta_prime;</div>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno">  153</span>            <span class="keywordflow">if</span> (x &lt;= 0.0)</div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno">  154</span>                <span class="keywordflow">throw</span> std::invalid_argument(<span class="stringliteral">&quot;[cholesky_decomposition::update] update makes matrix indefinite, no update available&quot;</span>);</div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno">  155</span>            <span class="keywordtype">double</span> nLjj = std::sqrt(x);</div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno">  156</span>            L(j,j) = nLjj;</div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno">  157</span>            beta_prime += swj2/dj;</div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno">  158</span>            </div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno">  159</span>            <span class="comment">// Update the terms of L</span></div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno">  160</span>            <span class="keywordflow">if</span>(j+1 &lt;n){</div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno">  161</span>                subrange(column(L,j),j+1,n) *= a;</div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno">  162</span>                noalias(subrange(temp,j+1,n)) -= (wj/Ljj) * subrange(column(L,j),j+1,n);</div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno">  163</span>                <span class="keywordflow">if</span>(gamma == 0)</div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno">  164</span>                    <span class="keywordflow">continue</span>;</div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno">  165</span>                subrange(column(L,j),j+1,n) *= nLjj/Ljj;</div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno">  166</span>                noalias(subrange(column(L,j),j+1,n))+= (nLjj * beta*wj/gamma)*subrange(temp,j+1,n);</div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno">  167</span>            }</div>
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno">  168</span>        }</div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno">  169</span>    }</div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno">  170</span>    </div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno">  171</span>    <span class="keyword">template</span>&lt;<span class="keyword">typename</span> Archive&gt;</div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno">  172</span>    <span class="keywordtype">void</span> serialize( Archive &amp; ar, <span class="keyword">const</span> std::size_t version ) {</div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno">  173</span>        ar &amp; m_cholesky;</div>
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno">  174</span>    }</div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno">  175</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno">  176</span>    MatrixStorage m_cholesky;</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno">  177</span>};</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno">  178</span> </div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno">  179</span><span class="comment"></span> </div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno">  180</span><span class="comment">/// \brief Symmetric eigenvalue decomposition A=QDQ^T</span></div>
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno">  181</span><span class="comment">///</span></div>
<div class="line"><a id="l00182" name="l00182"></a><span class="lineno">  182</span><span class="comment">/// every symmetric matrix can be decomposed into its eigenvalue decomposition</span></div>
<div class="line"><a id="l00183" name="l00183"></a><span class="lineno">  183</span><span class="comment">/// A=QDQ^T, where Q is an orthogonal matrix with Q^TQ=QQ^T=I</span></div>
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno">  184</span><span class="comment">/// and D is the diagonal matrix of eigenvalues of A.</span></div>
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno">  185</span><span class="comment"></span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatrixStorage&gt;</div>
<div class="line"><a id="l00186" name="l00186"></a><span class="lineno">  186</span><span class="keyword">class </span>symm_eigenvalue_decomposition:</div>
<div class="line"><a id="l00187" name="l00187"></a><span class="lineno">  187</span>    <span class="keyword">public</span> solver_expression&lt;</div>
<div class="line"><a id="l00188" name="l00188"></a><span class="lineno">  188</span>        symm_eigenvalue_decomposition&lt;MatrixStorage&gt;, </div>
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno">  189</span>        typename MatrixStorage::device_type</div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno">  190</span>&gt;{</div>
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno">  191</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00192" name="l00192"></a><span class="lineno">  192</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixStorage::value_type value_type;</div>
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno">  193</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> vector_temporary&lt;MatrixStorage&gt;::type VectorStorage;</div>
<div class="line"><a id="l00194" name="l00194"></a><span class="lineno">  194</span><span class="keyword">public</span>:</div>
<div class="line"><a id="l00195" name="l00195"></a><span class="lineno">  195</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixStorage::device_type device_type;</div>
<div class="line"><a id="l00196" name="l00196"></a><span class="lineno">  196</span>    symm_eigenvalue_decomposition(){}</div>
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno">  197</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> E&gt;</div>
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno">  198</span>    symm_eigenvalue_decomposition(matrix_expression&lt;E,device_type&gt; <span class="keyword">const</span>&amp; e){</div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno">  199</span>        decompose(e);</div>
<div class="line"><a id="l00200" name="l00200"></a><span class="lineno">  200</span>    }</div>
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno">  201</span>    </div>
<div class="line"><a id="l00202" name="l00202"></a><span class="lineno">  202</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> E&gt;</div>
<div class="line"><a id="l00203" name="l00203"></a><span class="lineno">  203</span>    <span class="keywordtype">void</span> decompose(matrix_expression&lt;E,device_type&gt; <span class="keyword">const</span>&amp; e){</div>
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno">  204</span>        REMORA_SIZE_CHECK(e().size1() ==  e().size2());</div>
<div class="line"><a id="l00205" name="l00205"></a><span class="lineno">  205</span>        m_eigenvectors.resize(e().size1(),e().size1());</div>
<div class="line"><a id="l00206" name="l00206"></a><span class="lineno">  206</span>        m_eigenvalues.resize(e().size1());</div>
<div class="line"><a id="l00207" name="l00207"></a><span class="lineno">  207</span>        noalias(m_eigenvectors) = e;</div>
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno">  208</span> </div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno">  209</span>        kernels::syev(m_eigenvectors,m_eigenvalues);</div>
<div class="line"><a id="l00210" name="l00210"></a><span class="lineno">  210</span>    }</div>
<div class="line"><a id="l00211" name="l00211"></a><span class="lineno">  211</span>    </div>
<div class="line"><a id="l00212" name="l00212"></a><span class="lineno">  212</span>    MatrixStorage <span class="keyword">const</span>&amp; Q()<span class="keyword">const</span>{</div>
<div class="line"><a id="l00213" name="l00213"></a><span class="lineno">  213</span>        <span class="keywordflow">return</span> m_eigenvectors;</div>
<div class="line"><a id="l00214" name="l00214"></a><span class="lineno">  214</span>    }</div>
<div class="line"><a id="l00215" name="l00215"></a><span class="lineno">  215</span>    VectorStorage <span class="keyword">const</span>&amp; D()<span class="keyword">const</span>{</div>
<div class="line"><a id="l00216" name="l00216"></a><span class="lineno">  216</span>        <span class="keywordflow">return</span> m_eigenvalues;</div>
<div class="line"><a id="l00217" name="l00217"></a><span class="lineno">  217</span>    }</div>
<div class="line"><a id="l00218" name="l00218"></a><span class="lineno">  218</span>    </div>
<div class="line"><a id="l00219" name="l00219"></a><span class="lineno">  219</span>    </div>
<div class="line"><a id="l00220" name="l00220"></a><span class="lineno">  220</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00221" name="l00221"></a><span class="lineno">  221</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, left)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00222" name="l00222"></a><span class="lineno">  222</span>        B() = Q() % to_diagonal(elem_inv(D()))% trans(Q()) % B;</div>
<div class="line"><a id="l00223" name="l00223"></a><span class="lineno">  223</span>    }</div>
<div class="line"><a id="l00224" name="l00224"></a><span class="lineno">  224</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00225" name="l00225"></a><span class="lineno">  225</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, right)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00226" name="l00226"></a><span class="lineno">  226</span>        <span class="keyword">auto</span> transB = trans(B);</div>
<div class="line"><a id="l00227" name="l00227"></a><span class="lineno">  227</span>        solve(transB,left());</div>
<div class="line"><a id="l00228" name="l00228"></a><span class="lineno">  228</span>    }</div>
<div class="line"><a id="l00229" name="l00229"></a><span class="lineno">  229</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB&gt;</div>
<div class="line"><a id="l00230" name="l00230"></a><span class="lineno">  230</span>    <span class="keywordtype">void</span> solve(vector_expression&lt;VecB,device_type&gt;&amp;b, left)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00231" name="l00231"></a><span class="lineno">  231</span>        b() = Q() % safe_div(trans(Q()) % b,D() ,0.0);</div>
<div class="line"><a id="l00232" name="l00232"></a><span class="lineno">  232</span>    }</div>
<div class="line"><a id="l00233" name="l00233"></a><span class="lineno">  233</span>    </div>
<div class="line"><a id="l00234" name="l00234"></a><span class="lineno">  234</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB&gt;</div>
<div class="line"><a id="l00235" name="l00235"></a><span class="lineno">  235</span>    <span class="keywordtype">void</span> solve(vector_expression&lt;VecB,device_type&gt;&amp;b, right)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00236" name="l00236"></a><span class="lineno">  236</span>        solve(b,left());</div>
<div class="line"><a id="l00237" name="l00237"></a><span class="lineno">  237</span>    }</div>
<div class="line"><a id="l00238" name="l00238"></a><span class="lineno">  238</span>    </div>
<div class="line"><a id="l00239" name="l00239"></a><span class="lineno">  239</span>    <span class="keyword">template</span>&lt;<span class="keyword">typename</span> Archive&gt;</div>
<div class="line"><a id="l00240" name="l00240"></a><span class="lineno">  240</span>    <span class="keywordtype">void</span> serialize( Archive &amp; ar, <span class="keyword">const</span> std::size_t version ) {</div>
<div class="line"><a id="l00241" name="l00241"></a><span class="lineno">  241</span>        ar &amp; m_eigenvectors;</div>
<div class="line"><a id="l00242" name="l00242"></a><span class="lineno">  242</span>        ar &amp; m_eigenvalues;</div>
<div class="line"><a id="l00243" name="l00243"></a><span class="lineno">  243</span>    }</div>
<div class="line"><a id="l00244" name="l00244"></a><span class="lineno">  244</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00245" name="l00245"></a><span class="lineno">  245</span>    MatrixStorage m_eigenvectors;</div>
<div class="line"><a id="l00246" name="l00246"></a><span class="lineno">  246</span>    VectorStorage m_eigenvalues;</div>
<div class="line"><a id="l00247" name="l00247"></a><span class="lineno">  247</span>};</div>
<div class="line"><a id="l00248" name="l00248"></a><span class="lineno">  248</span> </div>
<div class="line"><a id="l00249" name="l00249"></a><span class="lineno">  249</span> </div>
<div class="line"><a id="l00250" name="l00250"></a><span class="lineno">  250</span> </div>
<div class="line"><a id="l00251" name="l00251"></a><span class="lineno">  251</span> </div>
<div class="line"><a id="l00252" name="l00252"></a><span class="lineno">  252</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatrixStorage&gt;</div>
<div class="line"><a id="l00253" name="l00253"></a><span class="lineno">  253</span><span class="keyword">class </span>pivoting_lu_decomposition:</div>
<div class="line"><a id="l00254" name="l00254"></a><span class="lineno">  254</span><span class="keyword">public</span> solver_expression&lt;</div>
<div class="line"><a id="l00255" name="l00255"></a><span class="lineno">  255</span>    pivoting_lu_decomposition&lt;MatrixStorage&gt;, </div>
<div class="line"><a id="l00256" name="l00256"></a><span class="lineno">  256</span>    typename MatrixStorage::device_type</div>
<div class="line"><a id="l00257" name="l00257"></a><span class="lineno">  257</span>&gt;{</div>
<div class="line"><a id="l00258" name="l00258"></a><span class="lineno">  258</span><span class="keyword">public</span>:</div>
<div class="line"><a id="l00259" name="l00259"></a><span class="lineno">  259</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixStorage::device_type device_type;</div>
<div class="line"><a id="l00260" name="l00260"></a><span class="lineno">  260</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> E&gt;</div>
<div class="line"><a id="l00261" name="l00261"></a><span class="lineno">  261</span>    pivoting_lu_decomposition(matrix_expression&lt;E,device_type&gt; <span class="keyword">const</span>&amp; e)</div>
<div class="line"><a id="l00262" name="l00262"></a><span class="lineno">  262</span>    :m_factor(e), m_permutation(e().size1()){</div>
<div class="line"><a id="l00263" name="l00263"></a><span class="lineno">  263</span>        kernels::getrf(m_factor,m_permutation);</div>
<div class="line"><a id="l00264" name="l00264"></a><span class="lineno">  264</span>    }</div>
<div class="line"><a id="l00265" name="l00265"></a><span class="lineno">  265</span>    </div>
<div class="line"><a id="l00266" name="l00266"></a><span class="lineno">  266</span>    MatrixStorage <span class="keyword">const</span>&amp; factor()<span class="keyword">const</span>{</div>
<div class="line"><a id="l00267" name="l00267"></a><span class="lineno">  267</span>        <span class="keywordflow">return</span> m_factor;</div>
<div class="line"><a id="l00268" name="l00268"></a><span class="lineno">  268</span>    }</div>
<div class="line"><a id="l00269" name="l00269"></a><span class="lineno">  269</span>    </div>
<div class="line"><a id="l00270" name="l00270"></a><span class="lineno">  270</span>    permutation_matrix <span class="keyword">const</span>&amp; permutation()<span class="keyword"> const</span>{</div>
<div class="line"><a id="l00271" name="l00271"></a><span class="lineno">  271</span>        <span class="keywordflow">return</span> m_permutation;</div>
<div class="line"><a id="l00272" name="l00272"></a><span class="lineno">  272</span>    }</div>
<div class="line"><a id="l00273" name="l00273"></a><span class="lineno">  273</span>    </div>
<div class="line"><a id="l00274" name="l00274"></a><span class="lineno">  274</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00275" name="l00275"></a><span class="lineno">  275</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, left)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00276" name="l00276"></a><span class="lineno">  276</span>        swap_rows(m_permutation,B);</div>
<div class="line"><a id="l00277" name="l00277"></a><span class="lineno">  277</span>        kernels::trsm&lt;unit_lower,left&gt;(m_factor,B);</div>
<div class="line"><a id="l00278" name="l00278"></a><span class="lineno">  278</span>        kernels::trsm&lt;upper,left&gt;(m_factor,B);</div>
<div class="line"><a id="l00279" name="l00279"></a><span class="lineno">  279</span>    }</div>
<div class="line"><a id="l00280" name="l00280"></a><span class="lineno">  280</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00281" name="l00281"></a><span class="lineno">  281</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, right)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00282" name="l00282"></a><span class="lineno">  282</span>        kernels::trsm&lt;upper,right&gt;(m_factor,B);</div>
<div class="line"><a id="l00283" name="l00283"></a><span class="lineno">  283</span>        kernels::trsm&lt;unit_lower,right&gt;(m_factor,B);</div>
<div class="line"><a id="l00284" name="l00284"></a><span class="lineno">  284</span>        swap_columns_inverted(m_permutation,B);</div>
<div class="line"><a id="l00285" name="l00285"></a><span class="lineno">  285</span>    }</div>
<div class="line"><a id="l00286" name="l00286"></a><span class="lineno">  286</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB&gt;</div>
<div class="line"><a id="l00287" name="l00287"></a><span class="lineno">  287</span>    <span class="keywordtype">void</span> solve(vector_expression&lt;VecB,device_type&gt;&amp; b, left)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00288" name="l00288"></a><span class="lineno">  288</span>        swap_rows(m_permutation,b);</div>
<div class="line"><a id="l00289" name="l00289"></a><span class="lineno">  289</span>        kernels::trsv&lt;unit_lower,left&gt;(m_factor,b);</div>
<div class="line"><a id="l00290" name="l00290"></a><span class="lineno">  290</span>        kernels::trsv&lt;upper,left&gt;(m_factor,b);</div>
<div class="line"><a id="l00291" name="l00291"></a><span class="lineno">  291</span>    }</div>
<div class="line"><a id="l00292" name="l00292"></a><span class="lineno">  292</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB&gt;</div>
<div class="line"><a id="l00293" name="l00293"></a><span class="lineno">  293</span>    <span class="keywordtype">void</span> solve(vector_expression&lt;VecB,device_type&gt;&amp; b, right)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00294" name="l00294"></a><span class="lineno">  294</span>        kernels::trsv&lt;upper,right&gt;(m_factor,b);</div>
<div class="line"><a id="l00295" name="l00295"></a><span class="lineno">  295</span>        kernels::trsv&lt;unit_lower,right&gt;(m_factor,b);</div>
<div class="line"><a id="l00296" name="l00296"></a><span class="lineno">  296</span>        swap_rows_inverted(m_permutation,b);</div>
<div class="line"><a id="l00297" name="l00297"></a><span class="lineno">  297</span>    }</div>
<div class="line"><a id="l00298" name="l00298"></a><span class="lineno">  298</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00299" name="l00299"></a><span class="lineno">  299</span>    MatrixStorage m_factor;</div>
<div class="line"><a id="l00300" name="l00300"></a><span class="lineno">  300</span>    permutation_matrix m_permutation;</div>
<div class="line"><a id="l00301" name="l00301"></a><span class="lineno">  301</span>};</div>
<div class="line"><a id="l00302" name="l00302"></a><span class="lineno">  302</span> </div>
<div class="line"><a id="l00303" name="l00303"></a><span class="lineno">  303</span> </div>
<div class="line"><a id="l00304" name="l00304"></a><span class="lineno">  304</span><span class="comment">// This is an implementation suggested by</span></div>
<div class="line"><a id="l00305" name="l00305"></a><span class="lineno">  305</span><span class="comment">// &quot;Fast Computation of Moore-Penrose Inverse Matrices&quot;</span></div>
<div class="line"><a id="l00306" name="l00306"></a><span class="lineno">  306</span><span class="comment">// applied to the special case of symmetric pos semi-def matrices</span></div>
<div class="line"><a id="l00307" name="l00307"></a><span class="lineno">  307</span><span class="comment">// trading numerical accuracy vs speed. We go for speed.</span></div>
<div class="line"><a id="l00308" name="l00308"></a><span class="lineno">  308</span><span class="comment">//</span></div>
<div class="line"><a id="l00309" name="l00309"></a><span class="lineno">  309</span><span class="comment">// The fact that A is not full rank means it is not invertable,</span></div>
<div class="line"><a id="l00310" name="l00310"></a><span class="lineno">  310</span><span class="comment">// so we solve it in a least squares sense.</span></div>
<div class="line"><a id="l00311" name="l00311"></a><span class="lineno">  311</span><span class="comment">//</span></div>
<div class="line"><a id="l00312" name="l00312"></a><span class="lineno">  312</span><span class="comment">// We use the formula for the pseudo-inverse:</span></div>
<div class="line"><a id="l00313" name="l00313"></a><span class="lineno">  313</span><span class="comment">// (P^T A P)^-1 = L(L^TL)^-1(L^TL)^-1 L^T</span></div>
<div class="line"><a id="l00314" name="l00314"></a><span class="lineno">  314</span><span class="comment">// where L is a matrix obtained by some rank revealing factorization</span></div>
<div class="line"><a id="l00315" name="l00315"></a><span class="lineno">  315</span><span class="comment">// P^T A P = L L^T </span></div>
<div class="line"><a id="l00316" name="l00316"></a><span class="lineno">  316</span><span class="comment">// we chose a pivoting cholesky to make use of the fact that A is symmetric</span></div>
<div class="line"><a id="l00317" name="l00317"></a><span class="lineno">  317</span><span class="comment">// and all eigenvalues are &gt;=0. If A has full rank, this reduces to</span></div>
<div class="line"><a id="l00318" name="l00318"></a><span class="lineno">  318</span><span class="comment">// the cholesky factor where the pivoting leads to slightly smaller numerical errors</span></div>
<div class="line"><a id="l00319" name="l00319"></a><span class="lineno">  319</span><span class="comment">// At a higher computational cost compared to the normal cholesky decomposition.</span></div>
<div class="line"><a id="l00320" name="l00320"></a><span class="lineno">  320</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatrixStorage&gt;</div>
<div class="line"><a id="l00321" name="l00321"></a><span class="lineno">  321</span><span class="keyword">class </span>symm_pos_semi_definite_solver:</div>
<div class="line"><a id="l00322" name="l00322"></a><span class="lineno">  322</span>    <span class="keyword">public</span> solver_expression&lt;symm_pos_semi_definite_solver&lt;MatrixStorage&gt;, typename MatrixStorage::device_type&gt;{</div>
<div class="line"><a id="l00323" name="l00323"></a><span class="lineno">  323</span><span class="keyword">public</span>:</div>
<div class="line"><a id="l00324" name="l00324"></a><span class="lineno">  324</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixStorage::device_type device_type;</div>
<div class="line"><a id="l00325" name="l00325"></a><span class="lineno">  325</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> E&gt;</div>
<div class="line"><a id="l00326" name="l00326"></a><span class="lineno">  326</span>    symm_pos_semi_definite_solver(matrix_expression&lt;E,device_type&gt; <span class="keyword">const</span>&amp; e)</div>
<div class="line"><a id="l00327" name="l00327"></a><span class="lineno">  327</span>    :m_factor(e), m_permutation(e().size1()){</div>
<div class="line"><a id="l00328" name="l00328"></a><span class="lineno">  328</span>        m_rank = kernels::pstrf&lt;lower&gt;(m_factor,m_permutation);</div>
<div class="line"><a id="l00329" name="l00329"></a><span class="lineno">  329</span>        <span class="keywordflow">if</span>(m_rank == e().size1()) <span class="keywordflow">return</span>; <span class="comment">//full rank, so m_factor is lower triangular and we are done</span></div>
<div class="line"><a id="l00330" name="l00330"></a><span class="lineno">  330</span>        </div>
<div class="line"><a id="l00331" name="l00331"></a><span class="lineno">  331</span>        <span class="keyword">auto</span> L = columns(m_factor,0,m_rank);</div>
<div class="line"><a id="l00332" name="l00332"></a><span class="lineno">  332</span>        m_cholesky.decompose(prod(trans(L),L));</div>
<div class="line"><a id="l00333" name="l00333"></a><span class="lineno">  333</span>    }</div>
<div class="line"><a id="l00334" name="l00334"></a><span class="lineno">  334</span>    </div>
<div class="line"><a id="l00335" name="l00335"></a><span class="lineno">  335</span>    std::size_t rank()<span class="keyword">const</span>{</div>
<div class="line"><a id="l00336" name="l00336"></a><span class="lineno">  336</span>        <span class="keywordflow">return</span> m_rank;</div>
<div class="line"><a id="l00337" name="l00337"></a><span class="lineno">  337</span>    }</div>
<div class="line"><a id="l00338" name="l00338"></a><span class="lineno">  338</span>    </div>
<div class="line"><a id="l00339" name="l00339"></a><span class="lineno">  339</span>    <span class="comment">//compute C so that A^dagger = CC^T</span></div>
<div class="line"><a id="l00340" name="l00340"></a><span class="lineno">  340</span>    <span class="comment">//where A^dagger is the moore-penrose inverse</span></div>
<div class="line"><a id="l00341" name="l00341"></a><span class="lineno">  341</span>    <span class="comment">// m must be of size rank x n</span></div>
<div class="line"><a id="l00342" name="l00342"></a><span class="lineno">  342</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> Mat&gt;</div>
<div class="line"><a id="l00343" name="l00343"></a><span class="lineno">  343</span>    <span class="keywordtype">void</span> compute_inverse_factor(matrix_expression&lt;Mat,device_type&gt;&amp; C)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00344" name="l00344"></a><span class="lineno">  344</span>        REMORA_SIZE_CHECK(C().size1() == m_rank);</div>
<div class="line"><a id="l00345" name="l00345"></a><span class="lineno">  345</span>        REMORA_SIZE_CHECK(C().size2() == m_factor.size1());</div>
<div class="line"><a id="l00346" name="l00346"></a><span class="lineno">  346</span>        <span class="keywordflow">if</span>(m_rank == m_factor.size1()){<span class="comment">//matrix has full rank</span></div>
<div class="line"><a id="l00347" name="l00347"></a><span class="lineno">  347</span>            <span class="comment">//initialize as identity matrix and solve</span></div>
<div class="line"><a id="l00348" name="l00348"></a><span class="lineno">  348</span>            noalias(C) = identity_matrix&lt;double&gt;( m_factor.size1());</div>
<div class="line"><a id="l00349" name="l00349"></a><span class="lineno">  349</span>            swap_columns_inverted(m_permutation,C);</div>
<div class="line"><a id="l00350" name="l00350"></a><span class="lineno">  350</span>            kernels::trsm&lt;lower,left&gt;(m_factor,C);</div>
<div class="line"><a id="l00351" name="l00351"></a><span class="lineno">  351</span>        }<span class="keywordflow">else</span>{</div>
<div class="line"><a id="l00352" name="l00352"></a><span class="lineno">  352</span>            <span class="keyword">auto</span> L = columns(m_factor,0,m_rank);</div>
<div class="line"><a id="l00353" name="l00353"></a><span class="lineno">  353</span>            noalias(C) = trans(L);</div>
<div class="line"><a id="l00354" name="l00354"></a><span class="lineno">  354</span>            m_cholesky.solve(C,left());</div>
<div class="line"><a id="l00355" name="l00355"></a><span class="lineno">  355</span>            swap_columns_inverted(m_permutation,C);</div>
<div class="line"><a id="l00356" name="l00356"></a><span class="lineno">  356</span>        }</div>
<div class="line"><a id="l00357" name="l00357"></a><span class="lineno">  357</span>        </div>
<div class="line"><a id="l00358" name="l00358"></a><span class="lineno">  358</span>    }</div>
<div class="line"><a id="l00359" name="l00359"></a><span class="lineno">  359</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00360" name="l00360"></a><span class="lineno">  360</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, left)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00361" name="l00361"></a><span class="lineno">  361</span>        swap_rows(m_permutation,B);</div>
<div class="line"><a id="l00362" name="l00362"></a><span class="lineno">  362</span>        <span class="keywordflow">if</span>(m_rank == 0){<span class="comment">//matrix is zero</span></div>
<div class="line"><a id="l00363" name="l00363"></a><span class="lineno">  363</span>            B().clear();</div>
<div class="line"><a id="l00364" name="l00364"></a><span class="lineno">  364</span>        }<span class="keywordflow">else</span> <span class="keywordflow">if</span>(m_rank == m_factor.size1()){<span class="comment">//matrix has full rank</span></div>
<div class="line"><a id="l00365" name="l00365"></a><span class="lineno">  365</span>            kernels::trsm&lt;lower,left&gt;(m_factor,B);</div>
<div class="line"><a id="l00366" name="l00366"></a><span class="lineno">  366</span>            kernels::trsm&lt;upper,left&gt;(trans(m_factor),B);</div>
<div class="line"><a id="l00367" name="l00367"></a><span class="lineno">  367</span>        }<span class="keywordflow">else</span>{<span class="comment">//matrix is missing rank</span></div>
<div class="line"><a id="l00368" name="l00368"></a><span class="lineno">  368</span>            <span class="keyword">auto</span> L = columns(m_factor,0,m_rank);</div>
<div class="line"><a id="l00369" name="l00369"></a><span class="lineno">  369</span>            <span class="keyword">auto</span> Z =  eval_block(prod(trans(L),B));</div>
<div class="line"><a id="l00370" name="l00370"></a><span class="lineno">  370</span>            m_cholesky.solve(Z,left()); </div>
<div class="line"><a id="l00371" name="l00371"></a><span class="lineno">  371</span>            m_cholesky.solve(Z,left()); </div>
<div class="line"><a id="l00372" name="l00372"></a><span class="lineno">  372</span>            noalias(B) = prod(L,Z);</div>
<div class="line"><a id="l00373" name="l00373"></a><span class="lineno">  373</span>        }</div>
<div class="line"><a id="l00374" name="l00374"></a><span class="lineno">  374</span>        swap_rows_inverted(m_permutation,B);</div>
<div class="line"><a id="l00375" name="l00375"></a><span class="lineno">  375</span>    }</div>
<div class="line"><a id="l00376" name="l00376"></a><span class="lineno">  376</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00377" name="l00377"></a><span class="lineno">  377</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp; B, right)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00378" name="l00378"></a><span class="lineno">  378</span>        <span class="comment">//compute using symmetry of the system of equations</span></div>
<div class="line"><a id="l00379" name="l00379"></a><span class="lineno">  379</span>        <span class="keyword">auto</span> transB = trans(B);</div>
<div class="line"><a id="l00380" name="l00380"></a><span class="lineno">  380</span>        solve(transB, left());</div>
<div class="line"><a id="l00381" name="l00381"></a><span class="lineno">  381</span>    }</div>
<div class="line"><a id="l00382" name="l00382"></a><span class="lineno">  382</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB, <span class="keywordtype">bool</span> Left&gt;</div>
<div class="line"><a id="l00383" name="l00383"></a><span class="lineno">  383</span>    <span class="keywordtype">void</span> solve(vector_expression&lt;VecB,device_type&gt;&amp; b, system_tag&lt;Left&gt;)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00384" name="l00384"></a><span class="lineno">  384</span>        swap_rows(m_permutation,b);</div>
<div class="line"><a id="l00385" name="l00385"></a><span class="lineno">  385</span>        <span class="keywordflow">if</span>(m_rank == 0){<span class="comment">//matrix is zero</span></div>
<div class="line"><a id="l00386" name="l00386"></a><span class="lineno">  386</span>            b().clear();</div>
<div class="line"><a id="l00387" name="l00387"></a><span class="lineno">  387</span>        }<span class="keywordflow">else</span> <span class="keywordflow">if</span>(m_rank == m_factor.size1()){<span class="comment">//matrix has full rank</span></div>
<div class="line"><a id="l00388" name="l00388"></a><span class="lineno">  388</span>            kernels::trsv&lt;lower,left &gt;(m_factor,b);</div>
<div class="line"><a id="l00389" name="l00389"></a><span class="lineno">  389</span>            kernels::trsv&lt;upper,left &gt;(trans(m_factor),b);</div>
<div class="line"><a id="l00390" name="l00390"></a><span class="lineno">  390</span>        }<span class="keywordflow">else</span>{<span class="comment">//matrix is missing rank</span></div>
<div class="line"><a id="l00391" name="l00391"></a><span class="lineno">  391</span>            <span class="keyword">auto</span> L = columns(m_factor,0,m_rank);</div>
<div class="line"><a id="l00392" name="l00392"></a><span class="lineno">  392</span>            <span class="keyword">auto</span> z =  eval_block(prod(trans(L),b));</div>
<div class="line"><a id="l00393" name="l00393"></a><span class="lineno">  393</span>            m_cholesky.solve(z,left()); </div>
<div class="line"><a id="l00394" name="l00394"></a><span class="lineno">  394</span>            m_cholesky.solve(z,left()); </div>
<div class="line"><a id="l00395" name="l00395"></a><span class="lineno">  395</span>            noalias(b) = prod(L,z);</div>
<div class="line"><a id="l00396" name="l00396"></a><span class="lineno">  396</span>        }</div>
<div class="line"><a id="l00397" name="l00397"></a><span class="lineno">  397</span>        swap_rows_inverted(m_permutation,b);</div>
<div class="line"><a id="l00398" name="l00398"></a><span class="lineno">  398</span>    }</div>
<div class="line"><a id="l00399" name="l00399"></a><span class="lineno">  399</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00400" name="l00400"></a><span class="lineno">  400</span>    std::size_t m_rank;</div>
<div class="line"><a id="l00401" name="l00401"></a><span class="lineno">  401</span>    MatrixStorage m_factor;</div>
<div class="line"><a id="l00402" name="l00402"></a><span class="lineno">  402</span>    cholesky_decomposition&lt;MatrixStorage&gt; m_cholesky;</div>
<div class="line"><a id="l00403" name="l00403"></a><span class="lineno">  403</span>    permutation_matrix m_permutation;</div>
<div class="line"><a id="l00404" name="l00404"></a><span class="lineno">  404</span>};</div>
<div class="line"><a id="l00405" name="l00405"></a><span class="lineno">  405</span> </div>
<div class="line"><a id="l00406" name="l00406"></a><span class="lineno">  406</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA&gt;</div>
<div class="line"><a id="l00407" name="l00407"></a><span class="lineno">  407</span><span class="keyword">class </span>cg_solver:<span class="keyword">public</span> solver_expression&lt;cg_solver&lt;MatA&gt;, typename MatA::device_type&gt;{</div>
<div class="line"><a id="l00408" name="l00408"></a><span class="lineno">  408</span><span class="keyword">public</span>:</div>
<div class="line"><a id="l00409" name="l00409"></a><span class="lineno">  409</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatA::const_closure_type matrix_closure_type;</div>
<div class="line"><a id="l00410" name="l00410"></a><span class="lineno">  410</span>    <span class="keyword">typedef</span> <span class="keyword">typename</span> MatA::device_type device_type;</div>
<div class="line"><a id="l00411" name="l00411"></a><span class="lineno">  411</span>    cg_solver(matrix_closure_type <span class="keyword">const</span>&amp; e, <span class="keywordtype">double</span> epsilon = 1.e-10, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> max_iterations = 0)</div>
<div class="line"><a id="l00412" name="l00412"></a><span class="lineno">  412</span>    :m_expression(e), m_epsilon(epsilon), m_max_iterations(max_iterations){}</div>
<div class="line"><a id="l00413" name="l00413"></a><span class="lineno">  413</span>    </div>
<div class="line"><a id="l00414" name="l00414"></a><span class="lineno">  414</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00415" name="l00415"></a><span class="lineno">  415</span>    <span class="keywordtype">void</span> solve(</div>
<div class="line"><a id="l00416" name="l00416"></a><span class="lineno">  416</span>        matrix_expression&lt;MatB,device_type&gt;&amp; B, left,</div>
<div class="line"><a id="l00417" name="l00417"></a><span class="lineno">  417</span>        <span class="keywordtype">double</span> epsilon, <span class="keywordtype">unsigned</span> max_iterations</div>
<div class="line"><a id="l00418" name="l00418"></a><span class="lineno">  418</span>    )<span class="keyword">const</span>{</div>
<div class="line"><a id="l00419" name="l00419"></a><span class="lineno">  419</span>        <span class="keyword">typename</span> matrix_temporary&lt;MatB&gt;::type X = B;</div>
<div class="line"><a id="l00420" name="l00420"></a><span class="lineno">  420</span>        cg(m_expression,X, B, epsilon, max_iterations);</div>
<div class="line"><a id="l00421" name="l00421"></a><span class="lineno">  421</span>        noalias(B) = X;</div>
<div class="line"><a id="l00422" name="l00422"></a><span class="lineno">  422</span>    }</div>
<div class="line"><a id="l00423" name="l00423"></a><span class="lineno">  423</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB&gt;</div>
<div class="line"><a id="l00424" name="l00424"></a><span class="lineno">  424</span>    <span class="keywordtype">void</span> solve(</div>
<div class="line"><a id="l00425" name="l00425"></a><span class="lineno">  425</span>        matrix_expression&lt;MatB,device_type&gt;&amp; B, right,</div>
<div class="line"><a id="l00426" name="l00426"></a><span class="lineno">  426</span>        <span class="keywordtype">double</span> epsilon, <span class="keywordtype">unsigned</span> max_iterations</div>
<div class="line"><a id="l00427" name="l00427"></a><span class="lineno">  427</span>    )<span class="keyword">const</span>{</div>
<div class="line"><a id="l00428" name="l00428"></a><span class="lineno">  428</span>        <span class="keyword">auto</span> transB = trans(B);</div>
<div class="line"><a id="l00429" name="l00429"></a><span class="lineno">  429</span>        <span class="keyword">typename</span> transposed_matrix_temporary&lt;MatB&gt;::type X = transB;</div>
<div class="line"><a id="l00430" name="l00430"></a><span class="lineno">  430</span>        cg(m_expression,X,transB, epsilon, max_iterations);</div>
<div class="line"><a id="l00431" name="l00431"></a><span class="lineno">  431</span>        noalias(transB) = X;</div>
<div class="line"><a id="l00432" name="l00432"></a><span class="lineno">  432</span>    }</div>
<div class="line"><a id="l00433" name="l00433"></a><span class="lineno">  433</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB, <span class="keywordtype">bool</span> Left&gt;</div>
<div class="line"><a id="l00434" name="l00434"></a><span class="lineno">  434</span>    <span class="keywordtype">void</span> solve(</div>
<div class="line"><a id="l00435" name="l00435"></a><span class="lineno">  435</span>        vector_expression&lt;VecB,device_type&gt;&amp;b, system_tag&lt;Left&gt;, </div>
<div class="line"><a id="l00436" name="l00436"></a><span class="lineno">  436</span>        <span class="keywordtype">double</span> epsilon, <span class="keywordtype">unsigned</span> max_iterations</div>
<div class="line"><a id="l00437" name="l00437"></a><span class="lineno">  437</span>    )<span class="keyword">const</span>{</div>
<div class="line"><a id="l00438" name="l00438"></a><span class="lineno">  438</span>        <span class="keyword">typename</span> vector_temporary&lt;VecB&gt;::type x = b;</div>
<div class="line"><a id="l00439" name="l00439"></a><span class="lineno">  439</span>        cg(m_expression,x,b,epsilon,max_iterations);</div>
<div class="line"><a id="l00440" name="l00440"></a><span class="lineno">  440</span>        noalias(b) = x;</div>
<div class="line"><a id="l00441" name="l00441"></a><span class="lineno">  441</span>    }</div>
<div class="line"><a id="l00442" name="l00442"></a><span class="lineno">  442</span>    </div>
<div class="line"><a id="l00443" name="l00443"></a><span class="lineno">  443</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB, <span class="keywordtype">bool</span> Left&gt;</div>
<div class="line"><a id="l00444" name="l00444"></a><span class="lineno">  444</span>    <span class="keywordtype">void</span> solve(vector_expression&lt;VecB,device_type&gt;&amp;b, system_tag&lt;Left&gt; <a class="code hl_namespace" href="namespacetag.html">tag</a>)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00445" name="l00445"></a><span class="lineno">  445</span>        solve(b, <a class="code hl_namespace" href="namespacetag.html">tag</a>, m_epsilon, m_max_iterations);</div>
<div class="line"><a id="l00446" name="l00446"></a><span class="lineno">  446</span>    }</div>
<div class="line"><a id="l00447" name="l00447"></a><span class="lineno">  447</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB, <span class="keywordtype">bool</span> Left&gt;</div>
<div class="line"><a id="l00448" name="l00448"></a><span class="lineno">  448</span>    <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB,device_type&gt;&amp;B, system_tag&lt;Left&gt; <a class="code hl_namespace" href="namespacetag.html">tag</a>)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00449" name="l00449"></a><span class="lineno">  449</span>        solve(B, <a class="code hl_namespace" href="namespacetag.html">tag</a>, m_epsilon, m_max_iterations);</div>
<div class="line"><a id="l00450" name="l00450"></a><span class="lineno">  450</span>    }</div>
<div class="line"><a id="l00451" name="l00451"></a><span class="lineno">  451</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00452" name="l00452"></a><span class="lineno">  452</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatT, <span class="keyword">class</span> VecB, <span class="keyword">class</span> VecX&gt;</div>
<div class="line"><a id="l00453" name="l00453"></a><span class="lineno">  453</span>    <span class="keywordtype">void</span> cg(</div>
<div class="line"><a id="l00454" name="l00454"></a><span class="lineno">  454</span>        matrix_expression&lt;MatT, device_type&gt; <span class="keyword">const</span>&amp; A,</div>
<div class="line"><a id="l00455" name="l00455"></a><span class="lineno">  455</span>        vector_expression&lt;VecX, device_type&gt;&amp; x,</div>
<div class="line"><a id="l00456" name="l00456"></a><span class="lineno">  456</span>        vector_expression&lt;VecB, device_type&gt; <span class="keyword">const</span>&amp; b,</div>
<div class="line"><a id="l00457" name="l00457"></a><span class="lineno">  457</span>        <span class="keywordtype">double</span> epsilon,</div>
<div class="line"><a id="l00458" name="l00458"></a><span class="lineno">  458</span>        <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> max_iterations</div>
<div class="line"><a id="l00459" name="l00459"></a><span class="lineno">  459</span>    )<span class="keyword">const</span>{</div>
<div class="line"><a id="l00460" name="l00460"></a><span class="lineno">  460</span>        REMORA_SIZE_CHECK(A().size1() == A().size2());</div>
<div class="line"><a id="l00461" name="l00461"></a><span class="lineno">  461</span>        REMORA_SIZE_CHECK(A().size1() == b().size());</div>
<div class="line"><a id="l00462" name="l00462"></a><span class="lineno">  462</span>        REMORA_SIZE_CHECK(A().size1() == x().size());</div>
<div class="line"><a id="l00463" name="l00463"></a><span class="lineno">  463</span>        <span class="keyword">typedef</span> <span class="keyword">typename</span> vector_temporary&lt;VecX&gt;::type vector_type;</div>
<div class="line"><a id="l00464" name="l00464"></a><span class="lineno">  464</span>        </div>
<div class="line"><a id="l00465" name="l00465"></a><span class="lineno">  465</span>        std::size_t dim = b().size();</div>
<div class="line"><a id="l00466" name="l00466"></a><span class="lineno">  466</span>        </div>
<div class="line"><a id="l00467" name="l00467"></a><span class="lineno">  467</span>        <span class="comment">//initialize point. </span></div>
<div class="line"><a id="l00468" name="l00468"></a><span class="lineno">  468</span>        vector_type residual = b - prod(A,x);</div>
<div class="line"><a id="l00469" name="l00469"></a><span class="lineno">  469</span>        <span class="comment">//check if provided solution is better than starting at 0</span></div>
<div class="line"><a id="l00470" name="l00470"></a><span class="lineno">  470</span>        <span class="keywordflow">if</span>(norm_inf(residual) &gt; norm_inf(b)){</div>
<div class="line"><a id="l00471" name="l00471"></a><span class="lineno">  471</span>            x().clear();</div>
<div class="line"><a id="l00472" name="l00472"></a><span class="lineno">  472</span>            residual = b;</div>
<div class="line"><a id="l00473" name="l00473"></a><span class="lineno">  473</span>        }</div>
<div class="line"><a id="l00474" name="l00474"></a><span class="lineno">  474</span>        vector_type next_residual(dim); <span class="comment">//the next residual</span></div>
<div class="line"><a id="l00475" name="l00475"></a><span class="lineno">  475</span>        vector_type p = residual; <span class="comment">//the search direction- initially it is the gradient direction</span></div>
<div class="line"><a id="l00476" name="l00476"></a><span class="lineno">  476</span>        vector_type Ap(dim); <span class="comment">//stores prod(A,p)</span></div>
<div class="line"><a id="l00477" name="l00477"></a><span class="lineno">  477</span>        </div>
<div class="line"><a id="l00478" name="l00478"></a><span class="lineno">  478</span>        <span class="keywordflow">for</span>(std::size_t iter = 0;; ++iter){</div>
<div class="line"><a id="l00479" name="l00479"></a><span class="lineno">  479</span>            <span class="keywordflow">if</span>(max_iterations != 0 &amp;&amp; iter &gt;= max_iterations) <span class="keywordflow">break</span>;</div>
<div class="line"><a id="l00480" name="l00480"></a><span class="lineno">  480</span>            noalias(Ap) = prod(A,p);</div>
<div class="line"><a id="l00481" name="l00481"></a><span class="lineno">  481</span>            <span class="keywordtype">double</span> rsqr = norm_sqr(residual);</div>
<div class="line"><a id="l00482" name="l00482"></a><span class="lineno">  482</span>            <span class="keywordtype">double</span> alpha = rsqr/inner_prod(p,Ap);</div>
<div class="line"><a id="l00483" name="l00483"></a><span class="lineno">  483</span>            noalias(x) += alpha * p;</div>
<div class="line"><a id="l00484" name="l00484"></a><span class="lineno">  484</span>            noalias(next_residual) = residual - alpha * Ap; </div>
<div class="line"><a id="l00485" name="l00485"></a><span class="lineno">  485</span>            <span class="keywordflow">if</span>(norm_inf(next_residual) &lt; epsilon)</div>
<div class="line"><a id="l00486" name="l00486"></a><span class="lineno">  486</span>                <span class="keywordflow">break</span>;</div>
<div class="line"><a id="l00487" name="l00487"></a><span class="lineno">  487</span>            </div>
<div class="line"><a id="l00488" name="l00488"></a><span class="lineno">  488</span>            <span class="keywordtype">double</span> beta = inner_prod(next_residual,next_residual)/rsqr;</div>
<div class="line"><a id="l00489" name="l00489"></a><span class="lineno">  489</span>            p *= beta;</div>
<div class="line"><a id="l00490" name="l00490"></a><span class="lineno">  490</span>            noalias(p) += next_residual;</div>
<div class="line"><a id="l00491" name="l00491"></a><span class="lineno">  491</span>            <a class="code hl_function" href="namespaceshark.html#a3fffe112e8e09ea8f41e4fb7113e93ee" title="Swaps the contents of two instances of KeyValuePair.">swap</a>(residual,next_residual);</div>
<div class="line"><a id="l00492" name="l00492"></a><span class="lineno">  492</span>        }</div>
<div class="line"><a id="l00493" name="l00493"></a><span class="lineno">  493</span>    }</div>
<div class="line"><a id="l00494" name="l00494"></a><span class="lineno">  494</span>    </div>
<div class="line"><a id="l00495" name="l00495"></a><span class="lineno">  495</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatT, <span class="keyword">class</span> MatB, <span class="keyword">class</span> MatX&gt;</div>
<div class="line"><a id="l00496" name="l00496"></a><span class="lineno">  496</span>    <span class="keywordtype">void</span> cg(</div>
<div class="line"><a id="l00497" name="l00497"></a><span class="lineno">  497</span>        matrix_expression&lt;MatT, device_type&gt; <span class="keyword">const</span>&amp; A,</div>
<div class="line"><a id="l00498" name="l00498"></a><span class="lineno">  498</span>        matrix_expression&lt;MatX, device_type&gt;&amp; X,</div>
<div class="line"><a id="l00499" name="l00499"></a><span class="lineno">  499</span>        matrix_expression&lt;MatB, device_type&gt; <span class="keyword">const</span>&amp; B,</div>
<div class="line"><a id="l00500" name="l00500"></a><span class="lineno">  500</span>        <span class="keywordtype">double</span> epsilon,</div>
<div class="line"><a id="l00501" name="l00501"></a><span class="lineno">  501</span>        <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> max_iterations</div>
<div class="line"><a id="l00502" name="l00502"></a><span class="lineno">  502</span>    )<span class="keyword">const</span>{</div>
<div class="line"><a id="l00503" name="l00503"></a><span class="lineno">  503</span>        REMORA_SIZE_CHECK(A().size1() == A().size2());</div>
<div class="line"><a id="l00504" name="l00504"></a><span class="lineno">  504</span>        REMORA_SIZE_CHECK(A().size1() == B().size1());</div>
<div class="line"><a id="l00505" name="l00505"></a><span class="lineno">  505</span>        REMORA_SIZE_CHECK(A().size1() == X().size1());</div>
<div class="line"><a id="l00506" name="l00506"></a><span class="lineno">  506</span>        REMORA_SIZE_CHECK(B().size2() == X().size2());</div>
<div class="line"><a id="l00507" name="l00507"></a><span class="lineno">  507</span>        <span class="keyword">typedef</span> <span class="keyword">typename</span> vector_temporary&lt;MatX&gt;::type vector_type;</div>
<div class="line"><a id="l00508" name="l00508"></a><span class="lineno">  508</span>        <span class="keyword">typedef</span> <span class="keyword">typename</span> matrix_temporary&lt;MatX&gt;::type matrix_type;</div>
<div class="line"><a id="l00509" name="l00509"></a><span class="lineno">  509</span>        </div>
<div class="line"><a id="l00510" name="l00510"></a><span class="lineno">  510</span>        std::size_t dim = B().size1();</div>
<div class="line"><a id="l00511" name="l00511"></a><span class="lineno">  511</span>        std::size_t num_rhs = B().size2();</div>
<div class="line"><a id="l00512" name="l00512"></a><span class="lineno">  512</span>        </div>
<div class="line"><a id="l00513" name="l00513"></a><span class="lineno">  513</span>        <span class="comment">//initialize gradient given the starting point</span></div>
<div class="line"><a id="l00514" name="l00514"></a><span class="lineno">  514</span>        matrix_type residual = B - prod(A,X);</div>
<div class="line"><a id="l00515" name="l00515"></a><span class="lineno">  515</span>        <span class="comment">//check for each rhs whether the starting point is better than starting from scratch</span></div>
<div class="line"><a id="l00516" name="l00516"></a><span class="lineno">  516</span>        <span class="keywordflow">for</span>(std::size_t i = 0; i != num_rhs; ++i){</div>
<div class="line"><a id="l00517" name="l00517"></a><span class="lineno">  517</span>            <span class="keywordflow">if</span>(norm_inf(column(residual,i)) &lt;= norm_inf(column(residual,i))){</div>
<div class="line"><a id="l00518" name="l00518"></a><span class="lineno">  518</span>                column(X,i).clear();</div>
<div class="line"><a id="l00519" name="l00519"></a><span class="lineno">  519</span>                noalias(column(residual,i)) = column(B,i);</div>
<div class="line"><a id="l00520" name="l00520"></a><span class="lineno">  520</span>            }</div>
<div class="line"><a id="l00521" name="l00521"></a><span class="lineno">  521</span>        }</div>
<div class="line"><a id="l00522" name="l00522"></a><span class="lineno">  522</span>        </div>
<div class="line"><a id="l00523" name="l00523"></a><span class="lineno">  523</span>        vector_type next_residual(dim); <span class="comment">//the next residual of a column</span></div>
<div class="line"><a id="l00524" name="l00524"></a><span class="lineno">  524</span>        matrix_type P = residual; <span class="comment">//the search direction- initially it is the gradient direction</span></div>
<div class="line"><a id="l00525" name="l00525"></a><span class="lineno">  525</span>        matrix_type AP(dim, num_rhs); <span class="comment">//stores prod(A,p)</span></div>
<div class="line"><a id="l00526" name="l00526"></a><span class="lineno">  526</span>        </div>
<div class="line"><a id="l00527" name="l00527"></a><span class="lineno">  527</span>        <span class="keywordflow">for</span>(std::size_t iter = 0;; ++iter){</div>
<div class="line"><a id="l00528" name="l00528"></a><span class="lineno">  528</span>            <span class="keywordflow">if</span>(max_iterations != 0 &amp;&amp; iter &gt;= max_iterations) <span class="keywordflow">break</span>;</div>
<div class="line"><a id="l00529" name="l00529"></a><span class="lineno">  529</span>            <span class="comment">//compute the product for all rhs at the same time</span></div>
<div class="line"><a id="l00530" name="l00530"></a><span class="lineno">  530</span>            noalias(AP) = prod(A,P);</div>
<div class="line"><a id="l00531" name="l00531"></a><span class="lineno">  531</span>            <span class="comment">//for each rhs apply a step of cg</span></div>
<div class="line"><a id="l00532" name="l00532"></a><span class="lineno">  532</span>            <span class="keywordflow">for</span>(std::size_t i = 0; i != num_rhs; ++i){</div>
<div class="line"><a id="l00533" name="l00533"></a><span class="lineno">  533</span>                <span class="keyword">auto</span> r = column(residual,i);</div>
<div class="line"><a id="l00534" name="l00534"></a><span class="lineno">  534</span>                <span class="comment">//skip this if we are done already</span></div>
<div class="line"><a id="l00535" name="l00535"></a><span class="lineno">  535</span>                <span class="comment">//otherwise we might run into numerical troubles later on</span></div>
<div class="line"><a id="l00536" name="l00536"></a><span class="lineno">  536</span>                <span class="keywordflow">if</span>(norm_inf(r) &lt; epsilon) <span class="keywordflow">continue</span>;</div>
<div class="line"><a id="l00537" name="l00537"></a><span class="lineno">  537</span>                </div>
<div class="line"><a id="l00538" name="l00538"></a><span class="lineno">  538</span>                <span class="keyword">auto</span> x = column(X,i);</div>
<div class="line"><a id="l00539" name="l00539"></a><span class="lineno">  539</span>                <span class="keyword">auto</span> p = column(P,i);</div>
<div class="line"><a id="l00540" name="l00540"></a><span class="lineno">  540</span>                <span class="keyword">auto</span> Ap = column(AP,i);</div>
<div class="line"><a id="l00541" name="l00541"></a><span class="lineno">  541</span>                <span class="keywordtype">double</span> rsqr = norm_sqr(r);</div>
<div class="line"><a id="l00542" name="l00542"></a><span class="lineno">  542</span>                <span class="keywordtype">double</span> alpha = rsqr/inner_prod(p,Ap);</div>
<div class="line"><a id="l00543" name="l00543"></a><span class="lineno">  543</span>                noalias(x) += alpha * p;</div>
<div class="line"><a id="l00544" name="l00544"></a><span class="lineno">  544</span>                noalias(next_residual) = r - alpha * Ap; </div>
<div class="line"><a id="l00545" name="l00545"></a><span class="lineno">  545</span>                <span class="keywordtype">double</span> beta = inner_prod(next_residual,next_residual)/rsqr;</div>
<div class="line"><a id="l00546" name="l00546"></a><span class="lineno">  546</span>                p *= beta;</div>
<div class="line"><a id="l00547" name="l00547"></a><span class="lineno">  547</span>                noalias(p) += next_residual;</div>
<div class="line"><a id="l00548" name="l00548"></a><span class="lineno">  548</span>                noalias(r) = next_residual;</div>
<div class="line"><a id="l00549" name="l00549"></a><span class="lineno">  549</span>            }</div>
<div class="line"><a id="l00550" name="l00550"></a><span class="lineno">  550</span>            <span class="comment">//if all solutions are within tolerance, we are done</span></div>
<div class="line"><a id="l00551" name="l00551"></a><span class="lineno">  551</span>            <span class="keywordflow">if</span>(max(abs(residual)) &lt; epsilon)</div>
<div class="line"><a id="l00552" name="l00552"></a><span class="lineno">  552</span>                <span class="keywordflow">break</span>;</div>
<div class="line"><a id="l00553" name="l00553"></a><span class="lineno">  553</span>        }</div>
<div class="line"><a id="l00554" name="l00554"></a><span class="lineno">  554</span>    }</div>
<div class="line"><a id="l00555" name="l00555"></a><span class="lineno">  555</span>    </div>
<div class="line"><a id="l00556" name="l00556"></a><span class="lineno">  556</span>    matrix_closure_type m_expression;</div>
<div class="line"><a id="l00557" name="l00557"></a><span class="lineno">  557</span>    <span class="keywordtype">double</span> m_epsilon;</div>
<div class="line"><a id="l00558" name="l00558"></a><span class="lineno">  558</span>    <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> m_max_iterations;</div>
<div class="line"><a id="l00559" name="l00559"></a><span class="lineno">  559</span>};</div>
<div class="line"><a id="l00560" name="l00560"></a><span class="lineno">  560</span> </div>
<div class="line"><a id="l00561" name="l00561"></a><span class="lineno">  561</span><span class="comment"></span> </div>
<div class="line"><a id="l00562" name="l00562"></a><span class="lineno">  562</span><span class="comment">/////////////////////////////////////////////////////////////////</span></div>
<div class="line"><a id="l00563" name="l00563"></a><span class="lineno">  563</span><span class="comment">////Traits connecting decompositions/solvers with tags</span></div>
<div class="line"><a id="l00564" name="l00564"></a><span class="lineno">  564</span><span class="comment">/////////////////////////////////////////////////////////////////</span></div>
<div class="line"><a id="l00565" name="l00565"></a><span class="lineno">  565</span><span class="comment"></span><span class="keyword">struct </span>symm_pos_def{ <span class="keyword">typedef</span> symm_pos_def transposed_orientation;};</div>
<div class="line"><a id="l00566" name="l00566"></a><span class="lineno">  566</span><span class="keyword">struct </span>symm_semi_pos_def{ <span class="keyword">typedef</span> symm_semi_pos_def transposed_orientation;};</div>
<div class="line"><a id="l00567" name="l00567"></a><span class="lineno">  567</span><span class="keyword">struct </span>indefinite_full_rank{ <span class="keyword">typedef</span> indefinite_full_rank transposed_orientation;};</div>
<div class="line"><a id="l00568" name="l00568"></a><span class="lineno">  568</span><span class="keyword">struct </span>conjugate_gradient{</div>
<div class="line"><a id="l00569" name="l00569"></a><span class="lineno">  569</span>    <span class="keyword">typedef</span> conjugate_gradient transposed_orientation;</div>
<div class="line"><a id="l00570" name="l00570"></a><span class="lineno">  570</span>    <span class="keywordtype">double</span> epsilon;</div>
<div class="line"><a id="l00571" name="l00571"></a><span class="lineno">  571</span>    <span class="keywordtype">unsigned</span> max_iterations;</div>
<div class="line"><a id="l00572" name="l00572"></a><span class="lineno">  572</span>    conjugate_gradient(<span class="keywordtype">double</span> epsilon = 1.e-10, <span class="keywordtype">unsigned</span> max_iterations = 0)</div>
<div class="line"><a id="l00573" name="l00573"></a><span class="lineno">  573</span>    :epsilon(epsilon), max_iterations(max_iterations){}</div>
<div class="line"><a id="l00574" name="l00574"></a><span class="lineno">  574</span>};</div>
<div class="line"><a id="l00575" name="l00575"></a><span class="lineno">  575</span> </div>
<div class="line"><a id="l00576" name="l00576"></a><span class="lineno">  576</span> </div>
<div class="line"><a id="l00577" name="l00577"></a><span class="lineno">  577</span><span class="keyword">namespace </span>detail{</div>
<div class="line"><a id="l00578" name="l00578"></a><span class="lineno">  578</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA, <span class="keyword">class</span> SolverTag&gt;</div>
<div class="line"><a id="l00579" name="l00579"></a><span class="lineno">  579</span><span class="keyword">struct </span>solver_traits;</div>
<div class="line"><a id="l00580" name="l00580"></a><span class="lineno">  580</span> </div>
<div class="line"><a id="l00581" name="l00581"></a><span class="lineno">  581</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA, <span class="keywordtype">bool</span> Upper, <span class="keywordtype">bool</span> Unit&gt;</div>
<div class="line"><a id="l00582" name="l00582"></a><span class="lineno">  582</span><span class="keyword">struct </span>solver_traits&lt;MatA,triangular_tag&lt;Upper,Unit&gt; &gt;{</div>
<div class="line"><a id="l00583" name="l00583"></a><span class="lineno">  583</span>    <span class="keyword">class </span>type: <span class="keyword">public</span> solver_expression&lt;type,typename MatA::device_type&gt;{</div>
<div class="line"><a id="l00584" name="l00584"></a><span class="lineno">  584</span>    <span class="keyword">public</span>:</div>
<div class="line"><a id="l00585" name="l00585"></a><span class="lineno">  585</span>        <span class="keyword">typedef</span> <span class="keyword">typename</span> MatA::const_closure_type matrix_closure_type;</div>
<div class="line"><a id="l00586" name="l00586"></a><span class="lineno">  586</span>        <span class="keyword">typedef</span> <span class="keyword">typename</span> MatA::device_type device_type;</div>
<div class="line"><a id="l00587" name="l00587"></a><span class="lineno">  587</span>        type(matrix_closure_type <span class="keyword">const</span>&amp; e, triangular_tag&lt;Upper,Unit&gt;):m_matrix(e){}</div>
<div class="line"><a id="l00588" name="l00588"></a><span class="lineno">  588</span>        </div>
<div class="line"><a id="l00589" name="l00589"></a><span class="lineno">  589</span>        <span class="keyword">template</span>&lt;<span class="keyword">class</span> MatB, <span class="keywordtype">bool</span> Left&gt;</div>
<div class="line"><a id="l00590" name="l00590"></a><span class="lineno">  590</span>        <span class="keywordtype">void</span> solve(matrix_expression&lt;MatB, device_type&gt;&amp; B, system_tag&lt;Left&gt; ){</div>
<div class="line"><a id="l00591" name="l00591"></a><span class="lineno">  591</span>            kernels::trsm&lt;triangular_tag&lt;Upper,Unit&gt;,system_tag&lt;Left&gt; &gt;(m_matrix,B);</div>
<div class="line"><a id="l00592" name="l00592"></a><span class="lineno">  592</span>        }</div>
<div class="line"><a id="l00593" name="l00593"></a><span class="lineno">  593</span>        <span class="keyword">template</span>&lt;<span class="keyword">class</span> VecB, <span class="keywordtype">bool</span> Left&gt;</div>
<div class="line"><a id="l00594" name="l00594"></a><span class="lineno">  594</span>        <span class="keywordtype">void</span> solve(vector_expression&lt;VecB, device_type&gt;&amp; b, system_tag&lt;Left&gt; ){</div>
<div class="line"><a id="l00595" name="l00595"></a><span class="lineno">  595</span>            kernels::trsv&lt;triangular_tag&lt;Upper,Unit&gt;,system_tag&lt;Left&gt; &gt;(m_matrix,b);</div>
<div class="line"><a id="l00596" name="l00596"></a><span class="lineno">  596</span>        }</div>
<div class="line"><a id="l00597" name="l00597"></a><span class="lineno">  597</span>    <span class="keyword">private</span>:</div>
<div class="line"><a id="l00598" name="l00598"></a><span class="lineno">  598</span>        matrix_closure_type m_matrix;</div>
<div class="line"><a id="l00599" name="l00599"></a><span class="lineno">  599</span>    };</div>
<div class="line"><a id="l00600" name="l00600"></a><span class="lineno">  600</span>};</div>
<div class="line"><a id="l00601" name="l00601"></a><span class="lineno">  601</span>    </div>
<div class="line"><a id="l00602" name="l00602"></a><span class="lineno">  602</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA&gt;</div>
<div class="line"><a id="l00603" name="l00603"></a><span class="lineno">  603</span><span class="keyword">struct </span>solver_traits&lt;MatA,symm_pos_def&gt;{</div>
<div class="line"><a id="l00604" name="l00604"></a><span class="lineno">  604</span>    <span class="keyword">struct </span>type : <span class="keyword">public</span> cholesky_decomposition&lt;typename matrix_temporary&lt;MatA&gt;::type&gt;{</div>
<div class="line"><a id="l00605" name="l00605"></a><span class="lineno">  605</span>        <span class="keyword">template</span>&lt;<span class="keyword">class</span> M&gt;</div>
<div class="line"><a id="l00606" name="l00606"></a><span class="lineno">  606</span>        type(M <span class="keyword">const</span>&amp; m, symm_pos_def)</div>
<div class="line"><a id="l00607" name="l00607"></a><span class="lineno">  607</span>        :cholesky_decomposition&lt;typename matrix_temporary&lt;MatA&gt;::type&gt;(m){}</div>
<div class="line"><a id="l00608" name="l00608"></a><span class="lineno">  608</span>    };</div>
<div class="line"><a id="l00609" name="l00609"></a><span class="lineno">  609</span>};</div>
<div class="line"><a id="l00610" name="l00610"></a><span class="lineno">  610</span> </div>
<div class="line"><a id="l00611" name="l00611"></a><span class="lineno">  611</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA&gt;</div>
<div class="line"><a id="l00612" name="l00612"></a><span class="lineno">  612</span><span class="keyword">struct </span>solver_traits&lt;MatA,indefinite_full_rank&gt;{</div>
<div class="line"><a id="l00613" name="l00613"></a><span class="lineno">  613</span>    <span class="keyword">struct </span>type : <span class="keyword">public</span> pivoting_lu_decomposition&lt;typename matrix_temporary&lt;MatA&gt;::type&gt;{</div>
<div class="line"><a id="l00614" name="l00614"></a><span class="lineno">  614</span>        <span class="keyword">template</span>&lt;<span class="keyword">class</span> M&gt;</div>
<div class="line"><a id="l00615" name="l00615"></a><span class="lineno">  615</span>        type(M <span class="keyword">const</span>&amp; m, indefinite_full_rank)</div>
<div class="line"><a id="l00616" name="l00616"></a><span class="lineno">  616</span>        :pivoting_lu_decomposition&lt;typename matrix_temporary&lt;MatA&gt;::type&gt;(m){}</div>
<div class="line"><a id="l00617" name="l00617"></a><span class="lineno">  617</span>    };</div>
<div class="line"><a id="l00618" name="l00618"></a><span class="lineno">  618</span>};</div>
<div class="line"><a id="l00619" name="l00619"></a><span class="lineno">  619</span> </div>
<div class="line"><a id="l00620" name="l00620"></a><span class="lineno">  620</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA&gt;</div>
<div class="line"><a id="l00621" name="l00621"></a><span class="lineno">  621</span><span class="keyword">struct </span>solver_traits&lt;MatA,symm_semi_pos_def&gt;{</div>
<div class="line"><a id="l00622" name="l00622"></a><span class="lineno">  622</span>    <span class="keyword">struct </span>type : <span class="keyword">public</span> symm_pos_semi_definite_solver&lt;typename matrix_temporary&lt;MatA&gt;::type&gt;{</div>
<div class="line"><a id="l00623" name="l00623"></a><span class="lineno">  623</span>        <span class="keyword">template</span>&lt;<span class="keyword">class</span> M&gt;</div>
<div class="line"><a id="l00624" name="l00624"></a><span class="lineno">  624</span>        type(M <span class="keyword">const</span>&amp; m, symm_semi_pos_def)</div>
<div class="line"><a id="l00625" name="l00625"></a><span class="lineno">  625</span>        :symm_pos_semi_definite_solver&lt;typename matrix_temporary&lt;MatA&gt;::type&gt;(m){}</div>
<div class="line"><a id="l00626" name="l00626"></a><span class="lineno">  626</span>    };</div>
<div class="line"><a id="l00627" name="l00627"></a><span class="lineno">  627</span>};</div>
<div class="line"><a id="l00628" name="l00628"></a><span class="lineno">  628</span> </div>
<div class="line"><a id="l00629" name="l00629"></a><span class="lineno">  629</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA&gt;</div>
<div class="line"><a id="l00630" name="l00630"></a><span class="lineno">  630</span><span class="keyword">struct </span>solver_traits&lt;MatA,conjugate_gradient&gt;{</div>
<div class="line"><a id="l00631" name="l00631"></a><span class="lineno">  631</span>    <span class="keyword">struct </span>type : <span class="keyword">public</span> cg_solver&lt;MatA&gt;{</div>
<div class="line"><a id="l00632" name="l00632"></a><span class="lineno">  632</span>        <span class="keyword">template</span>&lt;<span class="keyword">class</span> M&gt;</div>
<div class="line"><a id="l00633" name="l00633"></a><span class="lineno">  633</span>        type(M <span class="keyword">const</span>&amp; m, conjugate_gradient t):cg_solver&lt;MatA&gt;(m,t.epsilon,t.max_iterations){}</div>
<div class="line"><a id="l00634" name="l00634"></a><span class="lineno">  634</span>    };</div>
<div class="line"><a id="l00635" name="l00635"></a><span class="lineno">  635</span>};</div>
<div class="line"><a id="l00636" name="l00636"></a><span class="lineno">  636</span> </div>
<div class="line"><a id="l00637" name="l00637"></a><span class="lineno">  637</span>}</div>
<div class="line"><a id="l00638" name="l00638"></a><span class="lineno">  638</span> </div>
<div class="line"><a id="l00639" name="l00639"></a><span class="lineno">  639</span><span class="keyword">template</span>&lt;<span class="keyword">class</span> MatA, <span class="keyword">class</span> SolverType&gt;</div>
<div class="line"><a id="l00640" name="l00640"></a><span class="lineno">  640</span><span class="keyword">struct </span>solver:<span class="keyword">public</span> detail::solver_traits&lt;MatA,SolverType&gt;::type{</div>
<div class="line"><a id="l00641" name="l00641"></a><span class="lineno">  641</span>    <span class="keyword">template</span>&lt;<span class="keyword">class</span> M&gt;</div>
<div class="line"><a id="l00642" name="l00642"></a><span class="lineno">  642</span>    solver(M <span class="keyword">const</span>&amp; m, SolverType t = SolverType()): detail::solver_traits&lt;MatA,SolverType&gt;::type(m,t){}</div>
<div class="line"><a id="l00643" name="l00643"></a><span class="lineno">  643</span>};</div>
<div class="line"><a id="l00644" name="l00644"></a><span class="lineno">  644</span> </div>
<div class="line"><a id="l00645" name="l00645"></a><span class="lineno">  645</span> </div>
<div class="line"><a id="l00646" name="l00646"></a><span class="lineno">  646</span>}</div>
<div class="line"><a id="l00647" name="l00647"></a><span class="lineno">  647</span><span class="preprocessor">#endif</span></div>
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